An allpass filter can be defined as any filter having a gain of
at all frequencies (but typically different delays at different
frequencies).

It is well known that the series combination of a feedforward and
feedback comb filter (having equal delays) creates an allpass filter
when the feedforward coefficient is the negative of the feedback
coefficient.

This can be recognized as a digital filter in direct form II
[449]. Thus, the system of Fig.2.30 can be interpreted as
the series combination of a feedback comb filter (Fig.2.24) taking
to followed by a feedforward comb filter (Fig.2.23)
taking to . By the commutativity of LTI systems, we can
interchange the order to get

Substituting the right-hand side of the first equation above for
in the second equation yields more simply

(3.15)

This can be recognized as direct form I [449], which requires
delays instead of ; however, unlike direct-form II,
direct-form I cannot suffer from ``internal'' overflow--overflow can
happen only at the output.

The coefficient symbols and here have been chosen to
correspond to standard notation for the transfer function

The frequency response is obtained by setting
,
where denotes radian frequency and denotes the sampling
period in seconds [449]. For an allpass filter, the frequency
magnitude must be the same for all
.

An allpass filter is obtained when
, or, in the case
of real coefficients, when . To see this, let
. Then we have